I got the required k value, which was 3, but the answer said the inequality was k≥3, while I got k≤3.

The marker's notes states that many students were unable to recognise the necessary sign change. Can someone please explain why the signs need to change?

Thank you.

Something my tutor always told me was always consider what everything means physically. Like what does discriminant mean and how does it affect the roots of the polynomial? What is it's physical significance? That way of thinking will help with more conceptual questions or this one for example where you have to link the fact that the discriminant implies the nature of the roots, i.e. are they real or not which will allow you to visualize the polynomial.

Hey everyone,I know this may sound silly but with days before my maths exams, and after prepping extensively on the more difficult topics, I realised today that I forgot how to do questions with bearings (as in prelim trig). I know that I need to use sine and cosine rule etc. but I have forgotten how to actually find the angles!If anyone could give me a quick refresher that would be great!!

Also here is a question that I am stuck with:Sally starts on J and swims on a bearing of 105 deg for 5km out to L. She then changes direction and swims on a bearing of 045 deg for a further 16km. (i) find angle JLM(ii) find the distance JM that Sally will swim back to J

Always a good idea to draw a diagram; that should always be your first step. I've put it in the spoiler below so this post takes up less space, but please take a look at it, it will go a long way to helping you understand any bearing questions that come your way

Spoiler

I've actually forgotten to put in the 45 degree angle, but from the information given in the question and angle sum of a triangle, you should be able to deduce that angle JLM = 120 degrees. And for part b), it's just one application of the cosine rule, with sides 5 and 16, with enclosed angle 120 degrees as found in part a). Have a go at more questions - starting with drawing a diagram! While you might be able to visualise it well, exams aren't really the places to hedge bets, it's better to draw them out always

Hi, I just wanted to know the difference between standard deviation and variance? I know that standard deviation measures the spread of data from the mean, so 68% of the data will be in 1 standard deviation and 95% of the data will be in 2 standard deviations. I was also told that variance is the same thing as standard deviation except its squared. So what's the point of having variance and how is it actually different to standard deviation and how is it useful in maths?

Hi, I just wanted to know the difference between standard deviation and variance? I know that standard deviation measures the spread of data from the mean, so 68% of the data will be in 1 standard deviation and 95% of the data will be in 2 standard deviations. I was also told that variance is the same thing as standard deviation except its squared. So what's the point of having variance and how is it actually different to standard deviation and how is it useful in maths?

Thanks for your time to answer my question

The standard deviation is nothing more but the square root of the variance.\[SD(X) = \sqrt{\operatorname{Var}(X)} \]Why do we have both? Because in the context of random variables, the absence of the square root makes the formula look nicer.\[ \operatorname{Var}(X) = E\left[(x-\mu)^2\right] \]That, and it's also more cleaner to use in more advanced (university level) mathematical statistics proofs.

But once the results are derived, it's usually more of interest to report standard deviations to the public. The variance rescales things according to square of quantities, whilst the standard deviation is on the same scale.

Edit: Upon looking at the question again, I guess the main answer is that the variance is more commonly used for proofs in mathematical statistics at university. The variance was the first convention used by mathematicians, and a lot of the theory of statistics has been developed based off it. For many famous distributions, the variance generally takes a nicer form - the standard deviation introduces a square root out of nowhere. Sure, a square root isn't overly harmful or anything, but it's just nicer to not have it there altogether. Also, computers may run into precision errors when dealing with square roots, as opposed to just positive integer powers.

But for summary statistics, I'd probably say one would be crazy to only use the variance instead of the SD.

The standard deviation is nothing more but the square root of the variance.\[SD(X) = \sqrt{\operatorname{Var}(X)} \]Why do we have both? Because in the context of random variables, the absence of the square root makes the formula look nicer.\[ \operatorname{Var}(X) = E\left[(x-\mu)^2\right] \]That, and it's also more cleaner to use in more advanced (university level) mathematical statistics proofs.

But once the results are derived, it's usually more of interest to report standard deviations to the public. The variance rescales things according to square of quantities, whilst the standard deviation is on the same scale.

So pretty much the variance and standard deviation are the exact same thing, except that variance is used to make the formulas look nicer and working out more cleaner.

Rui one more thing, for the Expected value of binomial distribution E(X) = np and Variance Var(X) = np(1-p), do we learn how to derive these formulas in school, cause I havent really seen anything in the books and if not, can u please show me how to derive these formulas???

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